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isotropic

Isotropic state is a bipartite quantum state.

These states are separable for α ≤ 1/(d+1), but are otherwise entangled.

isotropic 

isotropic(dim: int, alpha: float) -> ndarray

Produce a isotropic state 1.

Returns the isotropic state with parameter alpha acting on (dim-by-dim)-dimensional space. The isotropic state has the following form

\[ \begin{equation} \rho_{\alpha} = \frac{1 - \alpha}{d^2} \mathbb{I} \otimes \mathbb{I} + \alpha |\psi_+ \rangle \langle \psi_+ | \in \mathbb{C}^d \otimes \mathbb{C}^2 \end{equation} \]

where \(|\psi_+ \rangle = \frac{1}{\sqrt{d}} \sum_j |j \rangle \otimes |j \rangle\) is the maximally entangled state.

Parameters:

  • dim (int) –

    The local dimension.

  • alpha (float) –

    The parameter of the isotropic state.

Returns:

  • ndarray –

    Isotropic state of dimension dim.

Examples:

To generate the isotropic state with parameter \(\alpha=1/2\), we can make the following call to |toqito⟩ as

from toqito.states import isotropic
print(isotropic(3, 1 / 2))

[[0.22222222 0.         0.         0.         0.16666667 0.         0.         0.         0.16666667]
 [0.         0.05555556 0.         0.         0.         0.         0.         0.         0.        ]
 [0.         0.         0.05555556 0.         0.         0.         0.         0.         0.        ]
 [0.         0.         0.         0.05555556 0.         0.         0.         0.         0.        ]
 [0.16666667 0.         0.         0.         0.22222222 0.         0.         0.         0.16666667]
 [0.         0.         0.         0.         0.         0.05555556 0.         0.         0.        ]
 [0.         0.         0.         0.         0.         0.         0.05555556 0.         0.        ]
 [0.         0.         0.         0.         0.         0.         0.         0.05555556 0.        ]
 [0.16666667 0.         0.         0.         0.16666667 0.         0.         0.         0.22222222]]

References

1 Horodecki, Michal and Horodecki, Pawel. Reduction criterion of separability and limits for a class of protocols of entanglement distillation. (1998).

Source code in toqito/states/isotropic.py 
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def isotropic(dim: int, alpha: float) -> np.ndarray:
    r"""Produce a isotropic state [@horodecki1998reduction].

    Returns the isotropic state with parameter `alpha` acting on (`dim`-by-`dim`)-dimensional space.
    The isotropic state has the following form

    \[
        \begin{equation}
            \rho_{\alpha} = \frac{1 - \alpha}{d^2} \mathbb{I} \otimes
            \mathbb{I} + \alpha |\psi_+ \rangle \langle \psi_+ | \in
            \mathbb{C}^d \otimes \mathbb{C}^2
        \end{equation}
    \]

    where \(|\psi_+ \rangle = \frac{1}{\sqrt{d}} \sum_j |j \rangle \otimes |j \rangle\) is the maximally entangled
    state.

    Args:
        dim: The local dimension.
        alpha: The parameter of the isotropic state.

    Returns:
        Isotropic state of dimension `dim`.

    Examples:
        To generate the isotropic state with parameter \(\alpha=1/2\), we can make the following call to
        `|toqito⟩` as

        ```python exec="1" source="above" result="text"
        from toqito.states import isotropic
        print(isotropic(3, 1 / 2))
        ```

    """
    psi = max_entangled(dim, False, False)
    return (1 - alpha) * np.identity(dim**2) / dim**2 + alpha * psi @ psi.conj().T / dim